A proposal for dichroic experiments in the electron microscope

doi:10.1016/S0304-3991(03)00108-6

Ultramicroscopy

Volume 96, Issues 3–4, September 2003, Pages 463-468

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Abstract

Building upon the similarities between inelastic electron scattering and X-ray absorption we show that dichroism can be observed in electron energy loss spectrometry (EELS) in the transmission electron microscope (TEM). Natural or magnetic linear dichroism can be studied in electron scattering experiment with definite wave vector transfer in the interaction.

The detection of circular dichroism in the TEM relies on interferometric EELS in a particular scattering geometry that allows extraction of the mixed dynamic form factor from energy loss spectra. Similarities between dichroic signals in energy loss near edge structures and X-ray absorption near edge structures are discussed, and a new experimental setup for dichroic measurements in the TEM is proposed.

Introduction

Electron energy loss spectrometry (EELS) is a technique widely used for the local chemical analysis of materials in the transmission electron microscope (TEM) [1]. The fine structure of the ionisation edges (energy loss near edge structures (ELNES)) contains information about local electronic structure, bonding and oxidation state of the specimen in as much as the ELNES is given by a sum over site and angular-momentum-projected densities of states (DOS) above the Fermi level, multiplied by transition matrix elements.

It was shown that under dynamical diffraction conditions an interference term between the incident and the Bragg scattered beam occurs in the scattering cross section [2]. The same effect appears in electron compton scattering [3] and in the ELNES signal [4]. The interference term which is described by the mixed dynamic form factor of the scatterer relates to off-diagonal elements of the inelastically scattered probe electron's density matrix [5]. In scattering geometries invoking electron channelling the technique of “interferometric EELS” was applied to study dipole-forbidden transitions in the silicon L-edge in pure silicon [6], symmetry-selected final state orbitals in the O-K edge in TiO₂ (rutile) [7], and recently to detection of localisation in the plasmon [8].

The similarities between ELNES and X-ray absorption near edge structures (XANES) has long been recognised [9] but the two methods are still used in complementary ways. While ELNES gives easy access to light elements and relatively low-lying edges (below $2500 eV)$ , XANES is more suitable for the analysis of edges at high energy (up to more than $80 keV)$ . Nevertheless, there still is an energy range from 400 to $2500 eV$ where both methods are available. The energy resolution and brilliance of modern top-level analytical TEMs can compete with those of synchrotron. A TEM is extremely versatile, offering complementary analytical methods for crystallographic, electronic and chemical information on a nm or even sub-nm scale.

XANES analysis in the synchrotron using linearly or circularly polarised beams is widely used for the study of dichroism, particularly in magnetic materials, like magnetic d- and f-metals showing absorption edges with strong white lines. Dichroism is the dependence of the absorption cross section on the polarisation of the light, be that linear or circular. In linear dichroism, the important parameter is the orientation of the polarisation vector with respect to the specimen. Linear dichroism is simply related to the anisotropic structure of the material. Hexagonal structures like graphite are well-known examples. In circular dichroism, the absorption cross section depends on the helicity (right or left) of the incident photon. Natural circular dichroism is a rare effect requiring a crystal with specific structure, and was observed recently in LiIO₃ by Goulon et al. [10]. X-ray magnetic linear and circular dichroism (XMLD and XMCD) are similar effects but are driven by the distribution of magnetic moments in the sample [11]. The investigation of this phenomenon gained interest for the understanding of magnetism with the advent of synchrotrons and nearly all dichroic experiments deal with magnetic dichroism. Presently, important information on orbital and spin magnetisation in a variety of ferro-and ferrimagnetic compounds is being deduced from XMLD and XMCD with the aid of modern synchrotrons providing highly polarised beams with a brilliance of up to $10^{19} photon / s / mm^{2} / mrad^{2} /0.1%$ band width. In view of the similarity between X-ray absorption and inelastic electron scattering, the question arises whether or not it is possible to study dichroism in the TEM. For linear dichroism, the answer is positive; in this context, an experiment with a linearly polarised beam of photons is equivalent to an inelastic electron scattering experiment with definite wave vector transfer in the interaction. But what about circular dichroism? Does one need a circularly polarised beam of electrons? Is the electron spin the decisive quantity?

In the following we describe interferometric EELS, building upon expressions for the inelastic electron scattering cross section [12], [13]. This allows to demonstrate the similarities between electron scattering and X-ray absorption, and to show how linear and circular dichroism can be studied in ELNES experiments in the electron microscope.

Section snippets

Cross sections in EELS and XAS

The double differential scattering cross section for the transition from an initial state |i〉 to a final state $| f〉$ of a core electron to a final (empty) state by the interaction with a fast incoming electron that goes from state |ϕ_i〉 to state |ϕ_f〉 is given within the first-order Born approximation by $∂^{2} σ ∂ Ω ∂ E = 2π ℏ 1 j_{0} |〈ϕ_{i} |⊗〈i|V| f〉⊗|ϕ_{f} 〉|^{2} δν_{f} δ(E+E_{i} −E_{f}),$ where V is the Coulomb potential operator between the probing electron and the target charges and j₀ is the current density of the fast electron. When

Dipole approximation

If $q → · R → ⪡1$ then $e^{iq→·R→}$ can be expanded as $e^{iq→·R→} ≃1+ i q → · R →$

We can introduce this into , . In the case of ELNES, the first term of the expansion is the matrix element between initial and final states of the target electron and vanishes according to orthogonality. Eq. (2) becomes $∂^{2} σ ∂ E ∂ Ω = ∑ i, f 4γ^{2} a_{0}^{2} q^{4} k_{f} k_{i} |〈f| i q → · R → |i〉|^{2} δ(E_{i} −E_{f} +E).$

For XANES, the first term gives non-zero contribution and the second term can be neglected: $σ= ∑ i, f 4π^{2} ℏαω|〈f| ε → . R → |i〉|^{2} δ(E_{i} −E_{f} +E).$

Note that the notion “dipole” approximation

Linear dichroism

Natural linear dichroism is the orientation dependence of the interaction cross section in EELS and XANES in anisotropic samples. This is a well-known effect that can be understood by the fact that energy levels that are degenerate in a free atom are split in the solid (like the σ/π splitting in sp² hybridisation) and that orbitals corresponding to different energy levels are differently oriented in the lattice. Hexagonal structures like graphite or hexagonal boron nitride are perfect

Circular dichroism and MDFF

The magnetic circular dichroism relies on a coherent superposition of two linearly polarised photons with fixed phase difference (±π/2) $(ε → = ε →_{1} ± i ε →_{2})$ with $ε →_{1} ⊥ ε →_{2}$ . The normalised dichroic signal is defined by [11] $Δ γ(E) γ(E) = γ_{+} (E)−γ_{−} (E) γ_{+} (E)+γ_{−} (E)$ with γ₊₍₋₎(E) the absorption coefficient for right-(left-)polarised light. As such, the effect can be interpreted as an interference effect in the X-ray absorption cross section.

In the case where in EELS, the excitation arises from two coherent incident

Proposals for experimental setups

A possible experiment for detecting XMLD would be to measure the fine structure of a magnetic material in the diffraction plane with $q →$ pointing along the easy axis and perpendicular to it either by orienting the specimen in two directions, using double tilt goniometers or by making use of the method proposed by Botton and coworkers where the sample is tilted by 45° and the measurement made in diffraction mode on two points of the diffraction diagram [26]. In a polycrystalline material, one

Conclusion

We proposed a new experiment for XMCD in a TEM. The dichroic signal is expected to be of the same order of magnitude as in a synchrotron. This will allow to analyze specimen on a nm scale, but also the versatility of the TEM can be exploited in one and the same experiment, offering complementary analytical methods for crystallographic, electronic and chemical information on a nm or even sub-nm scale.

Acknowledgements

This work was supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung, project P14038-PHY.

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Ultramicroscopy

A proposal for dichroic experiments in the electron microscope

Abstract

Introduction

Section snippets

Cross sections in EELS and XAS

Dipole approximation

Linear dichroism

Circular dichroism and MDFF

Proposals for experimental setups

Conclusion

Acknowledgements

Ultramicroscopy

Adv. Electron. Electron Opt.

Micron

Ultramicroscopy

Electron Energy Loss Spectroscopy in the Electron Microscope

Philos. Mag. A

J. Phys. Condens. Matter

Philos. Mag. B

Phys. Rev. B

Phys. Rev. Lett.

Jpn. J. Appl. Phys.

J. Chem. Phys.

X-Ray Scattering and Absorption by Magnetic Materials